The subnormal embedding of complete groups
Theorem A: A finite soluble complete group A cannot be subnormally embedded in the derived subgroup of a finite group G. For \(A=S_ 3\), the symmetric group of degree 3, this is a result of P. Hall (\textit{R. S. Dark} [J. Lond. Math. Soc. 43, 387-390 (1968; Zbl 0157.054)]); this proof extends to the case where A is the holomorph of a cyclic 3-group. The general result is obtained from a structure theorem (Theorem C) for a directly indecomposable, finite, complete, subnormal subgroup A of a group G where A is not the holomorph of any cyclic 3-group: set \(A^*=ter\min al\) member of the lower central series of A, \(F=Fitting\) subgroup of \(A^ G\), then (a) \(A^*\) is normal in \(A^ G\), and \((A^*)^ G\) is the direct product of all conjugates of A, (b) AF is normal in \(A^ G\), and \(A^ G/F\) is the direct product of all conjugates of AF/F in G/F, (c) \(N_{G/F}(AF/F)=(AF/F)C_{G/F}(AF/F),\) (d) if G is finite, there is a supplement U of \(A^ G\) such that \(U\cap A^ G=F\).
scientific article; zbMATH DE number 3871629